I have since learned that TikiWiki includes forums and blogs, and that many features which require plugins in WordPress are included standard in TikiWiki.

I have also learned that WordPress can include forums and a wiki. I can incorporate blicki - a Wordpress plugin, to turn it into a wiki with revision histories and comparisons (I encoded "edit" into my templates), and I can add bbpress (a whole new application) along with the plugin "bbpress- integration", to make WordPress my one and only Content Management System.

Nonetheless, I have invested something like 1000 hours in developing the 3 sections, and if I move to use TikiWiki or WordPress exclusively, I imagine I would have to spend at least 100 hours to get it to the standards I have attained for my current setup.

I have set up OpenID for WordPress, and thus I have done half the work for setting up OpenID for TikiWiki and for phpBB, but still:
users would have to sign in to each section (using their OpenID), at least the first time.
So OpenID does not greatly help people who already have an OpenID, and for people who don't have an OpenID, it adds yet another step.
Beyond OpenID, how can I get single sign-on? (preferably without openID, but I will be happy even if it uses OpenID)

Even with single sign-on, searches and lists of categories, tags, and comments are not integrated.

Does Fantastico or anyone have a solution for that?

(Opinion: Fantastico can help, but since customers are invited to install multiple applications from a single source, without warning of these problems, customers assume the programs are at least somewhat cross-platform/integrated/interchangeable/ with connectivity, but instead Fantastico can cause more harm than good.)

None of this has to do with cPanel. Fantastico only assists in making a basic installation of some software easy for a new user to get started quickly. Having scripts such as these share database data has been accomplished in several communities that support the software, but you'll need to spend some time searching those community forums for tips in the right direction. You won't find that here though.

This forum is for server administrators that run cPanel software.

Scripts from Fantastico can be tied together (and upgraded via Fantastico) using bridging scripts and a little patience.